Research About The LCD Constacyclic Codes And LCD MDS Codes Based On Galois Inner Product

Galois inner product is an extension of Euclidean inner product and Hermitian inner product.Constacyclic codes are a kind of linear codes,which have a rich alge-braic structure and wide appplication.MDS codes have been described by renowned scholars MacWilliams and Sloane as the most enchanting error-correcting codes.L-CD codes axe widely used in datastorage,communication systems,electronics and cryptopraphy.In this thesis,we study the LCD constacyclic codes and LCD MDS codes based on the Galois inner product.We mainly discuss the usual form of Galois dual consta-cyclic codes and the necessary and sufficient conditions of Galois LCD constacyclic codes,and discuss three types of special Galois LCD MDS codes.Our main results are as follows:Let Fq be the finite field of ordex q,where q = pe,p is prime,e is positive integer,h E[0,e),where[0,e)= {0,1,…,e-1} is an integer interval.Rn,λ=Fq[x]/(xn-λ),λ∈Fq*.· Let C be[n,k]λ-consta-cyclic code,g(x)be a generator polynomial of C,xn-λ= g(x)h(x).Then C⊥h=<(h(x))λpe-k).When λ1+pe-h=1,C⊥h is also A-constacyclic code,then C is ph-LCD code if and only if(g(x))γpe-h=g(x),and all irreducible factor of g(x)have same multiplicity in g(x)and xn-λ.· Let q = pe,p be odd prime number.If there exists h,l ∈ {0,1,…,e},satisfied l le-h,e-h|e,then there exists a k-dimensional pl-LCD MDS code of length n = pl>3.· There exists Galois LCD MDS code of parameters[2e+2,3,2e],[2e+2,2e-1,4],[2k,k]and[2k + 1,k].